# The Arctic curve for Aztec rectangles with defects via the Tangent Method

## TL;DR

This paper applies the Tangent Method to derive the arctic curve for large Aztec rectangles with boundary defects, incorporating parameters that track specific steps and areas, thus extending understanding of domino tilings with irregular boundaries.

## Contribution

It introduces a novel application of the Tangent Method to Aztec rectangles with boundary defects, deriving the arctic curve for arbitrary defect distributions.

## Key findings

- Derived the arctic curve for Aztec rectangles with defects
- Extended the Tangent Method to irregular boundary conditions
- Provided examples illustrating different defect configurations

## Abstract

The Tangent Method of Colomo and Sportiello is applied to the study of the asymptotics of domino tilings of large Aztec rectangles, with some fixed distribution of defects along a boundary. The associated Non-Intersecting Lattice Path configurations are made of Schr\"oder paths whose weights involve two parameters $\gamma$ and $q$ keeping track respectively of one particular type of step and of the area below the paths. We derive the arctic curve for an arbitrary distribution of defects, and illustrate our result with a number of examples involving different classes of boundary defects.